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Properties

Label 275.2.y.a
Level $275$
Weight $2$
Character orbit 275.y
Analytic conductor $2.196$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,2,Mod(34,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.y (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{4} + 5 q^{5} + 4 q^{6} - 15 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 6 q^{4} + 5 q^{5} + 4 q^{6} - 15 q^{8} + 8 q^{9} - 10 q^{11} - 14 q^{14} - 20 q^{15} + 2 q^{16} - 20 q^{17} - 10 q^{19} + 10 q^{20} + 18 q^{21} + 10 q^{23} + 34 q^{24} + 5 q^{25} - 52 q^{26} - 15 q^{27} - 2 q^{29} + 55 q^{30} + 18 q^{31} - 24 q^{34} - 60 q^{35} - 36 q^{36} - 5 q^{37} - 20 q^{38} + 4 q^{39} + 14 q^{41} + 40 q^{42} + 6 q^{44} + 34 q^{46} - 25 q^{47} + 65 q^{48} + 40 q^{49} + 10 q^{50} - 10 q^{51} - 100 q^{52} + 5 q^{53} + 23 q^{54} + 14 q^{56} - 50 q^{58} - 31 q^{59} - 65 q^{60} + 40 q^{61} + 70 q^{62} + 20 q^{63} - 21 q^{64} - 10 q^{65} + 4 q^{66} - 60 q^{67} - 7 q^{69} + 20 q^{70} + 13 q^{71} - 5 q^{72} + 30 q^{73} - 6 q^{74} - 5 q^{75} - 80 q^{76} - 65 q^{78} + q^{79} + 75 q^{80} - 20 q^{83} - 32 q^{84} - 15 q^{85} - 13 q^{86} + 125 q^{87} + 30 q^{88} - 23 q^{89} + 45 q^{90} + 6 q^{91} - 60 q^{92} + 5 q^{94} + 30 q^{95} - 16 q^{96} - 15 q^{97} + 80 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −1.23778 + 1.70365i −1.64300 + 0.533844i −0.752312 − 2.31538i −0.803939 + 2.08655i 1.12419 − 3.45989i 3.34204i 0.870265 + 0.282766i −0.0125791 + 0.00913924i −2.55966 − 3.95232i
34.2 −1.12540 + 1.54898i 2.87194 − 0.933151i −0.514778 − 1.58432i −0.830554 − 2.07610i −1.78665 + 5.49874i − 2.96380i −0.608455 − 0.197699i 4.95024 − 3.59656i 4.15053 + 1.04993i
34.3 −0.834761 + 1.14895i 2.14368 − 0.696524i −0.00522576 − 0.0160832i 1.99483 + 1.01027i −0.989189 + 3.04441i 3.11054i −2.67850 − 0.870298i 1.68317 − 1.22289i −2.82596 + 1.44863i
34.4 −0.548729 + 0.755260i −1.41622 + 0.460159i 0.348719 + 1.07325i 0.336826 − 2.21055i 0.429583 − 1.32212i − 3.51969i −2.77766 − 0.902515i −0.633110 + 0.459982i 1.48472 + 1.46739i
34.5 −0.231037 + 0.317995i −2.57955 + 0.838146i 0.570291 + 1.75518i 2.22607 − 0.211241i 0.329445 − 1.01393i 2.49109i −1.43755 − 0.467087i 3.52453 − 2.56072i −0.447130 + 0.756683i
34.6 −0.192642 + 0.265150i 0.173097 − 0.0562428i 0.584841 + 1.79995i −2.23289 − 0.119194i −0.0184332 + 0.0567315i 1.82407i −1.21333 − 0.394234i −2.40025 + 1.74388i 0.461753 − 0.569088i
34.7 0.359798 − 0.495219i 0.205276 − 0.0666981i 0.502246 + 1.54576i 1.25473 + 1.85085i 0.0408275 − 0.125654i − 2.55437i 2.11053 + 0.685751i −2.38936 + 1.73597i 1.36803 + 0.0445649i
34.8 1.03040 − 1.41823i 2.05905 − 0.669025i −0.331608 − 1.02059i −0.323559 + 2.21253i 1.17282 − 3.60957i 0.209889i 1.54534 + 0.502112i 1.36503 − 0.991751i 2.80448 + 2.73869i
34.9 1.24872 − 1.71871i 0.0178328 − 0.00579423i −0.776643 − 2.39026i 1.29347 − 1.82399i 0.0123095 − 0.0378849i − 0.971667i −1.03705 − 0.336957i −2.42677 + 1.76315i −1.51972 − 4.50075i
34.10 1.53143 − 2.10783i −1.83210 + 0.595285i −1.47963 − 4.55384i −2.22401 + 0.231924i −1.55097 + 4.77338i − 0.968113i −6.90885 − 2.24482i 0.575174 − 0.417888i −2.91705 + 5.04300i
89.1 −1.23778 − 1.70365i −1.64300 − 0.533844i −0.752312 + 2.31538i −0.803939 − 2.08655i 1.12419 + 3.45989i − 3.34204i 0.870265 − 0.282766i −0.0125791 − 0.00913924i −2.55966 + 3.95232i
89.2 −1.12540 − 1.54898i 2.87194 + 0.933151i −0.514778 + 1.58432i −0.830554 + 2.07610i −1.78665 − 5.49874i 2.96380i −0.608455 + 0.197699i 4.95024 + 3.59656i 4.15053 − 1.04993i
89.3 −0.834761 − 1.14895i 2.14368 + 0.696524i −0.00522576 + 0.0160832i 1.99483 − 1.01027i −0.989189 − 3.04441i − 3.11054i −2.67850 + 0.870298i 1.68317 + 1.22289i −2.82596 − 1.44863i
89.4 −0.548729 − 0.755260i −1.41622 − 0.460159i 0.348719 − 1.07325i 0.336826 + 2.21055i 0.429583 + 1.32212i 3.51969i −2.77766 + 0.902515i −0.633110 − 0.459982i 1.48472 − 1.46739i
89.5 −0.231037 − 0.317995i −2.57955 − 0.838146i 0.570291 − 1.75518i 2.22607 + 0.211241i 0.329445 + 1.01393i − 2.49109i −1.43755 + 0.467087i 3.52453 + 2.56072i −0.447130 − 0.756683i
89.6 −0.192642 − 0.265150i 0.173097 + 0.0562428i 0.584841 − 1.79995i −2.23289 + 0.119194i −0.0184332 − 0.0567315i − 1.82407i −1.21333 + 0.394234i −2.40025 − 1.74388i 0.461753 + 0.569088i
89.7 0.359798 + 0.495219i 0.205276 + 0.0666981i 0.502246 − 1.54576i 1.25473 − 1.85085i 0.0408275 + 0.125654i 2.55437i 2.11053 − 0.685751i −2.38936 − 1.73597i 1.36803 − 0.0445649i
89.8 1.03040 + 1.41823i 2.05905 + 0.669025i −0.331608 + 1.02059i −0.323559 − 2.21253i 1.17282 + 3.60957i − 0.209889i 1.54534 − 0.502112i 1.36503 + 0.991751i 2.80448 − 2.73869i
89.9 1.24872 + 1.71871i 0.0178328 + 0.00579423i −0.776643 + 2.39026i 1.29347 + 1.82399i 0.0123095 + 0.0378849i 0.971667i −1.03705 + 0.336957i −2.42677 − 1.76315i −1.51972 + 4.50075i
89.10 1.53143 + 2.10783i −1.83210 − 0.595285i −1.47963 + 4.55384i −2.22401 − 0.231924i −1.55097 − 4.77338i 0.968113i −6.90885 + 2.24482i 0.575174 + 0.417888i −2.91705 − 5.04300i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.y.a ✓ 40
25.e even 10 1 inner 275.2.y.a ✓ 40
25.f odd 20 2 6875.2.a.m 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.y.a ✓ 40 1.a even 1 1 trivial
275.2.y.a ✓ 40 25.e even 10 1 inner
6875.2.a.m 40 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 13 T_{2}^{38} + 5 T_{2}^{37} + 113 T_{2}^{36} - 65 T_{2}^{35} - 820 T_{2}^{34} + 430 T_{2}^{33} + \cdots + 841 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display